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Anisotropic Complex Refractive Indices of Atomically Thin Materials: Determination of the Optical Constants of Few-Layer Black Phosphorus

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Anisotropic Complex Refractive Indices of Atomically Thin Materials: Determination of the Optical Constants of Few-Layer Black Phosphorus materials Article Anisotropic Complex Refractive Indices of Atomically Thin Materials: Determination of the Optical Constants of Few-Layer Black Phosphorus Aaron M. Ross 1, Giuseppe M. Paternò 2 , Stefano Dal Conte 1, Francesco Scotognella 1,2,* and Eugenio Cinquanta 3 1 Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy; [email protected] (A.M.R.); [email protected] (S.D.C.) 2 Center for Nano Science and Technology@PoliMi, Istituto Italiano di Tecnologia (IIT), Via Giovanni Pascoli, 70/3, 20133 Milan, Italy; [email protected] 3 Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche, 20133 Milan, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-02-2399-6056 Received: 5 November 2020; Accepted: 14 December 2020; Published: 16 December 2020 Abstract: In this work, studies of the optical constants of monolayer transition metal dichalcogenides and few-layer black phosphorus are briefly reviewed, with particular emphasis on the complex dielectric function and refractive index. Specifically, an estimate of the complex index of refraction of phosphorene and few-layer black phosphorus is given. The complex index of refraction of this material was extracted from differential reflectance data reported in the literature by employing a constrained Kramers–Kronig analysis combined with the transfer matrix method. The reflectance contrast of 1–3 layers of black phosphorus on a silicon dioxide/silicon substrate was then calculated using the extracted complex indices of refraction. Keywords: transition metal dichalcogenides; black phosphorus; two-dimensional materials; complex refractive index; Kramers–Kronig analysis 1. Introduction Two-dimensional materials emerged as a promising option for the development of nanoelectronic and optoelectronic devices, due to their peculiar electronic and optical properties. In this respect, transition metal dichalcogenides (TMDs), when thinned to a single atomic layer, show an indirect–direct band gap transition, inequivalent valleys in the Brillouin zone and non-trivial topological order [1–3]. In the vast family of two-dimensional materials, elemental ones represent a niche, due to their high reactivity at ambient conditions, that makes their characterization and exploitation challenging [4,5]. Among them, phosphorene, i.e., monolayer black phosphorous (BP), is the most promising material for optoelectronic and photonic applications, as its electronic band gap lies in a spectral region between the ones of TMDs and the one of graphene [6]. For an effective employment of these materials in photonic and optoelectronic devices, it is important to determine their dielectric response. Among the monolayers of transition metal dichalcogenides, MoS2 was the first material on which optical measurements were performed. In fact, in 2013 and 2014, the complex dielectric function of MoS2 was measured with spectroscopic ellipsometry by Shen et al. [7], Yim et al. [8] and Li et al. [9]. The complex dielectric functions of monolayers MoS2, WS2, MoSe2 and WSe2 were measured by Li et al. by employing reflectance contrast spectroscopy [10]. In 2019, the monolayers, together with the bilayers and the trilayers, of the four above mentioned chalcogenides were studied by Hsu et al. [11]. In 2020, Ermolaev et al. [12] measured the complex dielectric function of MoS2 in a broad range of Materials 2020, 13, 5736; doi:10.3390/ma13245736 www.mdpi.com/journal/materials Materials 2020, 13, 5736 2 of 12 wavelengths, i.e., 290–3300 nm. Ermolaev et al. [13] also measured the complex index of refraction of WS2 in the range of 375–1700 nm. Additionally, a number of studies investigated the anisotropic components of TMD and phosphorous few-layer systems via the incorporation of dielectric and plasmonic nanostructures [14–21]. In one study, the coupling of electromagnetic fields with the out-of-plane component of the MoS2 dielectric function was possible due to the peculiar morphology of chemical vapor deposition (CVD)-grown few-layer MoS2 on rippled substrate, hence promoting 1D nanostructures as a promising path for the full exploitation of the few-layer MoS2 dielectric response [19]. Similarly, a top-down approach was revealed to be effective for the strain-induced modification of the electronic band structure of atomically thin MoS2 and black phosphorous, paving the way for the on-demand tuning of their optical properties [20,21]. One study investigated exfoliated MoS2 on a lithographically defined SiO2 nanocone substrate, demonstrating a deterministic red-shift of the A exciton photoluminescence (PL) due to elastic strain [14]. Another study utilized spatially and time-resolved PL diffusion measurements of WSe2 exfoliated onto a 1.5 µm diameter SiO2 pillar, resulting in a similar red-shift and demonstration of exciton diffusion towards high tensile strain regions; that work was a proof of concept for engineerable excitonic diffusion that may lead to a new generation of opto-excitonic devices [15]. Strain engineering may also be used to enable new radiative pathways for the optical excitation of selection rule forbidden TMD dark excitons under normal incidence conditions; these dark and “gray” excitons possess much longer radiative lifetimes (>100 ps) [16,22,23]. In the absence of intentional strain, one study utilized a high numerical aperture objective lens (NA = 0.82) to characterize the dark exciton in WSe2: even for normal incidence, a Gaussian beam with a tight focus has a non-zero electric field component along the beam propagation direction [16]. However, optical excitation of the dark exciton in WSe2 was more easily enabled by coupling to the surface plasmon polariton of silver (Ag) [17]. Another study incorporated a hexagonal boron nitride (hBN)-encapsulated WSe2 monolayer into a plasmonic modulator device, requiring precise knowledge of the TMD, gold (Au) and hBN complex indices of refraction [18]. It thus emerges how the development of simple and robust tools for the careful analysis of the dispersion of the complex refractive index is beneficial for an appropriate engineering of devices that include two-dimensional materials. In contrast, phosphorene (single-layer BP) and few-layer BP have not been studied with the same fervor as their TMD counterparts, likely due to their relative sensitivity to degradation. One significant distinction between TMDs and BP is that the band structure of BP exhibits considerable anisotropy: that is, the effective electron mass along the armchair and zig-zag directions is a factor of five times larger in the former direction [24]. This anisotropy is manifest in polarization-sensitive absorption and PL measurements [24–27], as well as spatially resolved excitonic diffusion measurements [24]. The relatively fragile phosphorene system holds considerable promise due to the ease of PL/absorption tunability by layer number tuning, which can shift the resonance energies between 0.32 and 1.7 eV [24]. We focus on estimating the complex refractive index of phosphorene, which shows a morphological in-plane anisotropy resulting in an anisotropic optical response [25]. We retrieve the refractive index along the armchair (AC) and zig-zag (ZZ) direction by exploiting the constrained Kramers–Kronig analysis (CKKA) and fit experimental contrast reflectivity with a model obtained by means of the transfer matrix method (TMM). 2. Materials and Methods Methodology for extraction of the complex index of refraction of few-layer BP from reflectance contrast data: The linear optical properties of thin films are commonly measured via static absorption or reflectivity contrast measurements. Experimental techniques include normal incidence reflectivity contrast (dR/R) over a wide plane of view, spatially resolved confocal dR/R in the case of exfoliated small-area TMDs, few-layer BP, graphene and graphene oxide and simultaneous absolute reflectivity and transmission measurements for the determination of static absorption [10,11,25,27–30]. Although these methods reveal the static absorption A and reflectivity dR/R and transmission dT/T contrasts of the thin films Materials 2020, 13, x 5736 FOR PEER REVIEW 3 of 12 these methods reveal the static absorption A and reflectivity dR/R and transmission dT/T contrasts ofrelative the thin to the films linear relative substrate to the response, linear substrate the determination response, of the the determination complex index of of the refraction complexen =indexn + ofik, = + i n = p refractionor equivalently = the+ complex, or equivalently dielectric function the complex" dielectric"r "i, where function "=, typically +, where requires more= √, typicallysophisticated requires optical more characterization sophisticated methodsoptical char suchacterization as ellipsometry methods [8,31 such–33]. as ellipsometry [8,31– 33]. In this section, we extract the complex index of refraction of hBN-encapsulated few-layer BP on a sapphireIn this substratesection, we from extract confocal the complex reflectivity index contrast of refraction data taken of hBN-encapsulated from [25]. Specifically, few-layer the layered BP on sample structure is as follows (see Figure1): air 15 nm hBN few-layer BP α-Al O (sapphire); a sapphire substrate from confocal reflectivity contrast! data taken! from [25]. Specifically,! 2 3 the layered sampleBP thicknesses structure range is as follows nominally (see fromFigure 0.5 1): nmair (1L) 15 nm to 2.5 hBN nm (5L),few-layer and bulkBP BP -Al is2 100O3 (sapphire); nm thick. WeBP thicknesses assume that range the exfoliated nominally hBN from layer 0.5 is nm crystalline, (1L) to 2.5 rather nmthan (5L), isotropic and bulk as BP for is the 100 case nm of thick. cubic We BN assumeor polycrystalline that the exfoliated hBN [33– hBN35] with layer an is in-plane crystalline, dielectric rather constant than isotropic given byas for the case of cubic BN or polycrystalline hBN [33–35] with an in-plane dielectric constant given2 by 3.336 λ " = n2 = 1 +3.336 (1)(1) hBN, hBN, 2 ,∥ =k ,∥ = 1+ λ 26322 k − 26322− where the wavelength λ is given in nanometers [33]. Additionally, we also use a Sellmeier equation to where the wavelength is given in nanometers [33].
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